Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

Filter by
Sorted by
Tagged with
3 votes
0 answers
17 views

Confusion regarding vector/matrix multiplication in index notation

I came across this question and answer (sorry I don't have an electronic source for it, only a paper copy). After reading the answer it had me questioning the notation one uses to denote row/column ...
digital's user avatar
  • 145
0 votes
1 answer
63 views

$\det(A^2-B^2) \leq \det(A^2)$ when $B$ is of rank 1

This is an exercise problem given at linear algebra class. As a problem before this I was able to show that $$\det(A + uv^T) = \det A + v^T (adj A )u$$ when $A$ is order $n$ square matrix and $u,v$ ...
mathhello's user avatar
  • 902
0 votes
1 answer
23 views

Compute inverse of a special 2 by 2 block matrix.

Let $$X\in\mathbb{R}^p,\quad \tilde{X} = (1, X^{\top})^{\top}\in\mathbb{R}^{p+1},\quad \tilde{\Sigma}=\mathbb{E} \left[\tilde{X} \tilde{X}^{\top}\right]\in\mathbb{R}^{(p+1)\times (p+1)} $$ $$ (\tilde{...
maskeran's user avatar
  • 523
2 votes
1 answer
49 views

Find values of $a,b$. such that a matrix is diagonalizable

Find all values of $a,b\in\mathbb{R}$ such that $A$ is diagonalizable. $$A=\begin{pmatrix} -1 & a & b\\ 0 & 1 & 2\\ 0 & 2 & 1\\ \end{pmatrix}.$$ So far, I have that: $$det(A-\...
user926356's user avatar
  • 1,286
1 vote
0 answers
20 views

Reading $B^{-1}$ from simplex table

In uni I'm following a course on optimalisation and I have come across a problem. I am given the following minimalisation problem: and the corresponding final Simplex table: I now need to determine ...
Jord van Eldik's user avatar
0 votes
0 answers
17 views

3x3 real matrix decomposition to SVD using two unit quaternions and scale vector

I've been trying to search about doing 3x3 real matrix SVD, but instead of decomposing it into matrices, represent the two rotations as unit quaternions with the singular values as separate scale ...
Venom's user avatar
  • 1
-2 votes
0 answers
25 views

Further maths matrix question [closed]

The matrices p = [3/7, 3/7][1/7, 3/7] and Q = [11, 3][5,2] and R = [0,0][1,1] represent the transformations T, U and V respectively. a. A single transformation W is obtained by combining these ...
Peter Lux's user avatar
0 votes
0 answers
22 views

How to find the matrix $A$ from $C_0=A \times B$ or $C_1=B \times A$, given the singluar matrix B.

Kindly help me in the following: Let $C_0=A \times B$ and $C_1=B \times A$, where $A$ is a full rank square matrix, $B$ is a non-zero singular square matrix. Known $(B,C_0,C_1)$, how to calculate ...
X.H. Yue's user avatar
1 vote
0 answers
47 views

How to go about finding the minimal conditions guaranteeing that such a matrix is invertible?

I am working on an economic input-output model, and I want to find the conditions under which a system of linear equations determining the equilibrium yields a unique solution. I have an equation of ...
Balázs Markó's user avatar
3 votes
1 answer
21 views

Further explanation wanted on 'double Gaussian elimination' to triangularize a matrix.

I am trying to learn a more efficient way to triangularize a matrix. I found the following answer here on StackExchange which I found interesting, talking about 'double Gaussian elimination': Short ...
Newbie1000's user avatar
0 votes
2 answers
80 views

$n$ linear equations for $n+1$ unknowns

Let us consider a linear system \begin{eqnarray*} a_{11}x_1 + \dots + a_{1n}x_n + b_1y &=& c_1 \\ &\dots&\\ a_{n1}x_1 + \dots + a_{nn}x_n + b_ny &=& c_n \end{eqnarray*} for ...
Quiriacus's user avatar
-2 votes
0 answers
11 views

Using adjacency matrices to calculate graph rotations [closed]

Given the adjacency matrix $A$ of a tree $T$, is there a way of transforming that adjacency matrix in an efficient way to perform a rotation on that graph?
lo9ud's user avatar
  • 1
2 votes
0 answers
30 views

If $A$ and $B$ are normal with $\sigma(A),\sigma(B)\subseteq \mathbb{R}\cup\mathbb{T}$, does $\text{dim ker}(AB-BA)=\text{dim ker}(A^*B-BA^*)$?

I already asked this question for general $B$ and it was answered negatively here, so if the statement is true, one has to use the assumption on $B$ as well. In the original question I already ...
mathemagician99's user avatar
0 votes
0 answers
32 views

Block Matrix Inverse and Imaginary Number

I came here to ask some help regarding the following question. Let $M \in \mathbb R^{n \times n}$ be a positive semidefinite matrix, and $M^{(n)}$ is obtained from $M$ by zeroing all elements in the $...
jason 1's user avatar
  • 767
-3 votes
0 answers
56 views

Proving, if possible, that $A^2 = A$ implies $(I_3 - 2A) = (I_3 -2A)^{-1}‎$, for $A\in M_R(3)$ [closed]

Prove, if possible, that for $A\in M_R(3)$: $$A^2 = A \quad\implies\quad (I_3 - 2A) \;=\; (I_3 -2A)^{-1}‎$$
Angel B.'s user avatar
2 votes
0 answers
35 views

The Invariance of the Distribution of a Matrix

I'm currently stuck to the following statement. Given $W\in\mathbb R^{p\times d}$ having i.i.d entries from $N(0,1/d)$, define $Q\in\mathbb R^{p\times p}$ as follows: $$Q_{i,j} = \begin{cases}w_i^...
jason 1's user avatar
  • 767
2 votes
1 answer
13 views

Equation for Counting Unique RREF "Cases" for any (m x n) Matrix

Given a real matrix $A$ of size $(m \times n)$, I am seeking to determine the number of unique reduced row echelon form (RREF) "Cases" that exist for a given matrix size. Two RREF matrices ...
FaffyWaffles's user avatar
-1 votes
0 answers
18 views

SVD for finite and infinite dimensional matrix

Suppose I have a matrix $A_n$ that is symmetric and full rank, then I could apply eigenvalue decomposition on this matrix $A_n = U_n \Sigma_n U_n'$. Now suppose, the size of $A_n$ grows to infinity(We ...
Jerry's user avatar
  • 1
-2 votes
1 answer
48 views

Basis trick for Subspaces [closed]

If I have a vector space $V$ of dimension $4$ over the real numbers , and there is a subspace $U$ of $V$ of dimension $3$, then if I find $3$ linearly independent vectors, will they automatically span ...
adisnjo's user avatar
  • 255
1 vote
1 answer
16 views

How to derive the relation about Jordan decomposition of a matrix?

Assume that $v$ is an eigenvector with an eigenvalue of $0$ in matrix $H$, and its Jordan decomposition $H^J=SHS^{-1}$ satisfies $$ H^J=\left( \begin{matrix} 0& 1& \cdots& 0\\ 0& ...
SHBooKP's user avatar
  • 194
1 vote
1 answer
18 views

Positive definiteness of a diagonal matrix and boundedness of specific set

Problem: Let $D\in \mathbb R^{n\times n}$ be a diagonal matrix. Show that the set $$E=\{x\in \mathbb R^{n\times 1} \mid x^tDx\leq 1 \}$$ is bounded if and only if the diagonal entries of $D$ are ...
categoricallystupid's user avatar
2 votes
1 answer
73 views

Matrix representation of conic sections

The quadratic equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, i. e. the equation satisfied by conic sections, can be represented in the form $a^T Q a$, where $a$ is the column vector $$ \begin{...
Davide Masi's user avatar
-1 votes
1 answer
42 views

We know if $||A||$ is operator norm of $A$, then $||Ax|| \leq ||A||.||x||$ Is there any general rule when does it hold with equality for all $x$? [closed]

We know if $||A||$ is operator norm of $A$, then $||Ax|| \leq ||A||.||x||$ Is there any general rule when does it hold with equality for all $x$ For example: if $A$ is k$I$ for any scaler $k$, it ...
Debojjal Bagchi's user avatar
1 vote
0 answers
39 views

When are all Eigenvalues of a Matrix negative?

Suppose I have the following matrix where: All diagonals are negative In each row, the absolute value of each diagonal element is greater than the sum of all other element in that row Suppose this ...
stats_noob's user avatar
  • 3,172
-1 votes
0 answers
30 views

Is there a way to find the eigenvalues of a matrix using character table? [closed]

I am studying applications of representation theory. I want to know if there is a procedure to find the eigenvalues and eigenvectors of a matrix using the character table of the Group acting on its ...
Marisha Singh's user avatar
0 votes
1 answer
53 views

Sequences of nested triangles

Let $ T_0 = \triangle A_0B_0C_0 $ and $ \lbrace T_n = \triangle A_nB_nC_n\rbrace_{n\geq 0} $ be the sequence of nested triangles where $ T_{n+1} $ is formed by taking the midpoints of the sides of $ ...
math.enthusiast9's user avatar
1 vote
0 answers
21 views

$\|B\|_{op} \le \|A\|_{op}$ when $B_{i,j} = v_i^TAv_j$

I'm currently stuck to the following statement. Let $A\in\mathbb R^{n\times n}$ be a diagonal matrix whose diagonal elements are only 0 or 1, and let $B$ be $n\times n$ matrix such that $B_{i,j} = ...
jason 1's user avatar
  • 767
1 vote
1 answer
31 views

A question on positivity of eigenvalues for a matrix with some random chosen entries

Let $0<c<1$. Is it possible to construct a symmetric $n\times n$ matrix $A=(a_{ij})$ with $a_{ii}=1$ for all $1\leq i\leq n$ and $a_{ij}\in ${$1, c$} for all $i\neq j$ for some $n\geq 1$ such ...
ougao's user avatar
  • 3,683
-2 votes
0 answers
36 views

There is some ambiguity in kcet question [closed]

If A and B are two matrices such that AB = B and BA = A then A² + B² = ? (A) AB (B) A+B (C) 2BA (D) 2AB Option B,C,D is correct I guess but in answer key only option B is given My working
Dushyanth Yadav v's user avatar
2 votes
1 answer
22 views

Minimum columns of a matrix such that there is at least one nozero value in each row

I have a question that might belong better on the CS SE; let me know and I will move it. This comes from a research project and is a reframing of our question. Essentially, we have some matrix that is ...
regionalsky's user avatar
1 vote
0 answers
8 views

Comparison of Transformation Matrices U with Different Parameters

I want to transform any complex matrix into this U form. $$U = \begin{pmatrix} a & b \\\\ -b^* & a^* \end{pmatrix} = \begin{pmatrix} cos\theta exp(i\lambda) & sin\theta exp(i\mu)...
junghyunHa's user avatar
-4 votes
0 answers
52 views

Busted my head for 3 days looking for a solution, help pls anyone [closed]

i need to find the operator matrix F from a given operator matrix A ][1]
H.sandel's user avatar
2 votes
0 answers
30 views

Positive definite matrices and pivots.

I found these lecture notes online about positive definite matrices. To summarise what's needed for my question, they give 4 ways of working out if a matrix is positive definite. In particular, it ...
Mathematista's user avatar
2 votes
1 answer
44 views

Integer Linear Programming - Dividing n people into m groups

I have modeled the problem of dividing n people into m groups using a binary $nxn$ matrix that we will call X. If $x_{ij} = 1$ it means that person i is with person j in the solution's groups. If $x_{...
Zufra's user avatar
  • 185
-2 votes
0 answers
29 views

Does square root of matrices obey matrix inequalities? [closed]

Consider two positive semi-definite matrices $A,B$ such that $A\geq B \geq 0$. Then, is it true that: $A^{1/2} \geq B^{1/2}$?
SHUBHAYAN SARKAR's user avatar
1 vote
2 answers
45 views

Uniqueness of Hessenberg matrix in Cholesky factorization of Hankel matrix

I'm reading the paper how bad are Hankel matrices?, where the author claimed the following: Lemma 2.1. For any positive definite Hankel matrix $H \in \mathbb{R}^{n \times n}$ there exist a ...
wsz_fantasy's user avatar
  • 1,684
1 vote
1 answer
54 views

Proving a matrix is diagonalizable

Consider the matrix $M\in M_3(\mathbb{R})$ $$M=\begin{pmatrix} 5 & -6 & -6 \\ -1 & 4 & 2\\ 3 &-6&-4 \end{pmatrix}.$$ I need to show that $M$ is diagonalizable, and find a ...
user926356's user avatar
  • 1,286
2 votes
3 answers
310 views

Transform a column vector into its transpose with matrix multiplication? [closed]

I don't have tensor in mind here, what follows is just a question about linear algebra: If I start out with an $n\times 1$ column vector over the reals, are there any combinations of matrix ...
user avatar
0 votes
0 answers
18 views

For the following inner product space $V$ and $T ∈ L(V )$, evaluate$ T ^ ∗$ at a given point in $V$

Pardon for the loose title but I have this question: For the following inner product space $V$ and $T ∈ L(V )$, evaluate $T^*$ at a given point in $ V$: $V=P_1(R)$, with $\langle f(x),g(x) \rangle =\...
Kshitij Kumar's user avatar
1 vote
1 answer
41 views

Proving Row Independence in Matrix Product Resulting from Linearly Independent Vectors and Permutations

Let's denote by $u_1, u_2, \ldots, u_n$ a set of linearly independent vectors. For each $i$ from $1$ to $n$, we define $\sigma_i$ as distinct permutations of the numbers $1$ to $n$. Construct matrix $...
Fernand's user avatar
  • 15
0 votes
0 answers
12 views

If a singular square matrix is multiplied by a non-singular square matrix, the null space of the result is what?

If a non-singular square matrix $B$ is multiplied by a singular square matrix $A$ of the same order, the nullspace of the resulting matrix $C=B\times A$ or $C'=A \times B$ remains unchanged from that ...
X.H. Yue's user avatar
0 votes
1 answer
47 views

Let $C$ be a $4 \times 4$ matrix with all eigenvalues $λ = 2, −1$ and eigenspaces; calculate $C^4u$

Pardon for a poor title but I have this question which I am very confused: Let $C$ be a $4 \times 4$ matrix with all eigenvalues $λ = 2, −1$ and following eigenspaces: $E_2=\begin{pmatrix} 1\\1 \\ 1\\...
Kshitij Kumar's user avatar
0 votes
1 answer
22 views

Lower Bound for eigenvalue of a special form of symmetric tridiagonal matrix

Given the special tridiagonal matrix, where all constant $c_i >1$ and can be arbitrary large. \begin{equation} V^\top V = \begin{bmatrix} 2c_1^2 & -c_1 c_2 & 0 & \cdots &...
Nam Trần's user avatar
0 votes
1 answer
31 views

Any vector can be obtained by rotation and scaling from a unit vector?

How can i justify rigorously that for some vector $x$ there is a unitary matrix $R$ such that $x=\|x\|Ry$ where $y$ is some normed vector. This intuitively very clear to me but somewhat I have no idea ...
Perelman's user avatar
  • 219
0 votes
0 answers
12 views

Set of all Matrices generated via Addition/Subtraction and Scalar/Matrix Multiplication from Generator Set

Given a set of matrices over a field with algebraic closure, how can the set of matrices generated via addition/subtraction and scalar multiplication as well as matrix multiplication be characterised? ...
quantumorsch's user avatar
0 votes
0 answers
22 views

Most efficient way to recompute matrix cost after swapping columns

I work with regular matrices $M \in \mathbb N^{n\times n}$ with $n > 1$. My cost function is: $$\text{Cost} = \sum_{i=1, j>i}^n M_{ij}$$ Hence, we only consider the superior triangle, diagonal ...
Albert Schrödinberg's user avatar
0 votes
0 answers
15 views

singular matrix and number of eigenvectors

what is the relationship between the a singular matrix and the number of linearly independent eigenvectors? i encountered this question in DE system, and here the number of linearly independent ...
ZOOOOEE's user avatar
1 vote
2 answers
56 views

if the character of a g-representation has real values then it is realizable over the reals

Prove or disprove that if the character of a G-representation (where G is a finite group) has real values (i.e. $\chi_{\phi}(g)\in\mathbb{R}\forall g\in G$) then it is realizable over the reals (i.e. ...
user1127's user avatar
  • 471
1 vote
0 answers
29 views

Diagonalization of a block matrices

Suppose I have some $2^n$ by $2^n$ block matrix: $$\begin{pmatrix} A & B \\ B & A \end{pmatrix} $$ Where $A$ and $B$ are both $2^{n-1}$ by $2^{n-1}$ symmetric matrices with positive real ...
Jbag1212's user avatar
  • 1,455
0 votes
0 answers
7 views

Markov chain transition matrix optimal solution for Prime Climb

There is a game that involves a pawn starting at the number 0 in a field from 0 to 101. Its goal is to make it to 101. With each turn, I roll two dice. I can pick one of the two dice values and either ...
user1306253's user avatar

1
2 3 4 5
1120